Exam 8: Sequences, Series, and Probability

Use mathematical induction to prove the following inequality for all $n \geq 2$ . $\frac { 1 } { \sqrt { 23 } } + \frac { 1 } { \sqrt { 46 } } + \frac { 1 } { \sqrt { 69 } } + \ldots + \frac { 1 } { \sqrt { 23 n } } > \frac { \sqrt { n } } { \sqrt { 23 } }$

Find the specified $n$ th term in the expansion of the binomial. (Write the expansion in descending powers of $x$ .) $( 2 x + 3 y ) ^ { 6 } , n = 3$

Find the indicated partial sum of the series. $\sum _ { i = 1 } ^ { \infty } 5 \left( \frac { 1 } { 3 } \right) ^ { i }$ fourth partial sum A) $\frac { 5 } { 81 }$ B) $\frac { 605 } { 81 }$ C) $\frac { 200 } { 81 }$ D) $\frac { 65 } { 81 }$ E) $\frac { 205 } { 81 }$ Use mathematical induction to prove the property for all positive integers $n$ . $\left[ a ^ { n } \right] ^ { 3 } = a ^ { 3 n }$

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } 36 i ^ { 5 } = 3 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)$

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that $n$ begins with 1.) $a _ { n } = - \left( \frac { 1 } { 6 } \right) ^ { n }$

Write the first five terms of the geometric sequence. $a _ { 1 } = - 5 , r = - \frac { 1 } { 9 }$

Use mathematical induction to prove that 80 is a factor of $2 ^ { 8 n + 2 } + 16$ for all positive $n$ .

Find the sum of the integers from $- 34$ to $- 12$ .

Find the coefficient $a$ of the term in the expansion of the binomial. Binomial     $( 4 x + 3 y ) ^ { 6 }$ Term     $a x ^ { 2 } y ^ { 4 }$

Use summation notation to write the sum. $2 - 6 + 18 - \ldots - 486$

How many 3 -digit numbers can be formed if the leading digit cannot be zero and repeats are not allowed?

Find the sum. $\sum _ { i = 1 } ^ { 4 } ( - i - 4 )$

Find a formula for the $n t h$ term of the following geometric sequence, then find the 4 th term of the sequence. $7,28,112 , \ldots$

Given the sequence $4 + \frac { 12 } { 13 } , 4 + \frac { 19 } { 20 } , 4 + \frac { 26 } { 27 } , 4 + \frac { 33 } { 34 } , 4 + \frac { 40 } { 41 } , \ldots$ , write an expression for the apparent $n$ th term assuming $n$ begins with 1 .

Find the indicated $n$ th partial sum of the arithmetic sequence. $3.6,5.7,7.8,9.9 , \ldots , n = 60$

Write the first five terms of the sequence. (Assume that $n$ begins with 0 .) $a _ { n } = \frac { 3 ^ { n } } { ( n + 1 ) ! }$

Determine whether the sequence is geometric. If so, find the common ratio. $3,6,12,24 , \ldots$

Calculate the binomial coefficient: $\left( \begin{array} { c } 11 \\ 6 \end{array} \right)$